Halve squash and remove seeds. Cut into 2-inch sections and pare off rind. Cut flesh into thin julienne strips. Drop into boiling salted water; return to boil and then drain. Drop into a bowl of ice water, then drain. Spread on towel to dry.
Combine ginger, salt, lemon juice, and vinegar in a jar; shake to blend. Add oil and shake again. Combine three-quarters of the dressing with the squash. Toss and refrigerate until ready to serve.
To serve, toss remaining dressing with endive. Add squash and pomegranate seeds and mix gently.